A127384 -id:A127384 - cp's OEIS frontend

A127379 Signature-permutation of Callan's 2006 bijection on Dyck Paths, mirrored version (A057164-conjugate).

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 13, 12, 14, 15, 19, 22, 21, 16, 20, 17, 18, 23, 24, 25, 27, 26, 28, 29, 33, 36, 35, 30, 34, 31, 32, 37, 38, 39, 41, 40, 51, 52, 60, 64, 63, 56, 62, 58, 59, 42, 43, 53, 54, 55, 44, 61, 45, 46, 47, 57, 48, 50, 49, 65, 66, 67, 69, 68, 70, 71

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

It's much easier to implement Callan's 2006 bijection for S-expressions if one considers a mirror-image of the graphical description given by Callan (on page 3). Then this automorphism is just RIBS-transformation (explained in A122200) of the automorphism A127377 and Callan's original variant A127381 is obtained as A057164(a(A057164(n))).

A. Karttunen, Table of n, a(n) for n = 0..2055
D. Callan, A Bijection on Dyck Paths and Its Cycle Structure, 2006, 17pp.
Index entries for signature-permutations of Catalan automorphisms

Inverse: A127380. a(n) = A057164(A127381(A057164(n))). The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127384 and A086625 shifted once right. The maximum cycles and LCM's of cycle sizes begin as 1, 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, ... A127302(a(n)) = A127302(n) holds for all n. A127388 shows a variant which is an involution.

Differs from A073289 and A122349 for the first time at n=54, where a(n)=54, while A073289(54) = A122349(54) = 61.

A127381 Signature-permutation of Callan's 2006 bijection on Dyck Paths.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 13, 12, 14, 15, 19, 22, 18, 16, 21, 17, 20, 23, 24, 25, 27, 26, 28, 29, 33, 36, 32, 30, 35, 31, 34, 37, 38, 39, 41, 40, 51, 52, 60, 64, 46, 47, 59, 50, 49, 42, 43, 56, 63, 55, 44, 58, 45, 48, 53, 62, 54, 61, 57, 65, 66, 67, 69, 68, 70, 71

Offset: 0

Antti Karttunen, Jan 16 2007

This automorphism is much easier to implement for S-expressions when viewed as a A057164-conjugate of A127379. See the comments in the latter entry.

A. Karttunen, Table of n, a(n) for n = 0..2055
D. Callan, A Bijection on Dyck Paths and its Cycle Structure, The Electronic Journal of Combinatorics 14 (2007), #R28
Index entries for signature-permutations of Catalan automorphisms

Inverse: A127382. a(n) = A057164(A127379(A057164(n))). The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127384 and A086625 shifted once right. The maximum cycles and LCM's of cycle sizes begin as 1, 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, ...

A127383 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A127377/A127378.

Original entry on oeis.org

1, 1, 1, 2, 6, 15, 46, 141, 446, 1427, 4722, 15884, 54224, 187380

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

A127384, A127385, A127389.

A175137 Irregular triangle T(n,k) read by rows: number of orbits of size 2^k on Dyck n-paths.

Original entry on oeis.org

1, 2, 3, 1, 6, 2, 1, 12, 7, 4, 26, 23, 11, 2, 59, 71, 41, 8, 138, 224, 151, 30, 332, 709, 550, 114, 814, 2253, 1993, 406, 16, 2028, 7189, 7211, 1564, 64, 5118, 23045, 26221, 6010, 240, 13054, 74213, 95583, 23062, 912, 33598, 239979, 349145, 88530, 3504, 87143

Offset: 1

R. J. Mathar, Feb 21 2010

			Triangle starts at row n=1
1;
2;
3,1;
6,2,1;
12,7,4;
26,23,11,2;
59,71,41,8;
138,224,151,30;

David Callan, A bijection on Dyck paths and its cycle structure, El. J. Combinat. 14 (2007) # R28

Cf. A127384 (row sums).

Fx := proc(k) local ak ; ak := (2*x)^(2^k+1) ; (1-ak-(1-4*x+(ak*x*(2-ak))/(1-x))^(1/2))/(2*x-ak) ; end proc: ff := [] : for k from 0 to 5 do ff := [op(ff), taylor(Fx(k),x=0,18)] ; end do : F := proc(n,k) global ff ; coeftayl(op(k+1,ff),x=0,n) ; end proc: T := proc(n,k) global ff ; if k = 0 then F(n,0) ; else (F(n,k)-F(n,k-1))/2^k ; end if; end proc: for n from 1 to 17 do for k from 0 to 5 do if T(n,k) <> 0 then printf("%d,",T(n,k)) ; fi; end do ; printf("\n") ; end do ;

cp's OEIS Frontend

A127379 Signature-permutation of Callan's 2006 bijection on Dyck Paths, mirrored version (A057164-conjugate).

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A127381 Signature-permutation of Callan's 2006 bijection on Dyck Paths.

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A127383 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A127377/A127378.

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A175137 Irregular triangle T(n,k) read by rows: number of orbits of size 2^k on Dyck n-paths.

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